On Ribet’s lemma for GL\(_2\) modulo prime powers

Let \(\rho :G\rightarrow {{\,\textrm{GL}\,}}_2(K)\) $\rho :G\to {\text{GL}}_2(K)$ be a continuous representation of a compact group G over a complete discretely valued field K with ring of integers \(\mathcal {O}\) $\mathcal{O}$ and uniformiser \(\pi \) $\pi$ . We prove that \({{\,\textrm{tr}\,}}\rho \) $\text{tr}\rho$ is reducible modulo \(\pi ^n\) ${\pi }^n$ if and only if \(\rho \) $\rho$ is reducible modulo \(\pi ^n\) ${\pi }^n$ . More precisely, there exist characters \(\chi _1,\chi _2 :G\rightarrow (\mathcal {O}/\pi ^n\mathcal {O})^\times \) ${\chi }_1,{\chi }_2:G\to {(\mathcal{O}/{\pi }^n\mathcal{O})}^{\times }$ such that \(\det (t - \rho (g))\equiv (t-\chi _1(g))(t-\chi _2(g))\pmod {\pi ^n}\) $det(t-\rho (g))\equiv (t-{\chi }_1(g))(t-{\chi }_2(g))(mod{\pi }^n)$ for all \(g\in G\) $g\in G$ , if and only if there exists a G-stable lattice \(\Lambda \subseteq K^2\) $\mathrm{\Lambda }\subseteq K^2$ such that \(\Lambda /\pi ^n\Lambda \) $\mathrm{\Lambda }/{\pi }^n\mathrm{\Lambda }$ contains a G-invariant, free, rank one \(\mathcal {O}/\pi ^n\mathcal {O}\) $\mathcal{O}/{\pi }^n\mathcal{O}$ -submodule. Our result applies in the case that \(\rho \) $\rho$ is not residually multiplicity-free, in which case it answers a question of Bellaïche and Chenevier (J Algebra 410:501–525, 2014, pp. 524). As an application, we prove an optimal version of Ribet’s lemma, which gives a condition for the existence of a G-stable lattice \(\Lambda \) $\mathrm{\Lambda }$ that realises a non-split extension of \(\chi _2\) ${\chi }_2$ by \(\chi _1\) ${\chi }_1$ .